In the production of gears, especially bevel gears, two types of processes are commonly employed, generating processes and non-generating processes.
Generating processes can be divided into two categories, face milling (intermittent indexing) and face hobbing (continuous indexing). In generating face milling processes, a rotating tool is fed into the workpiece to a predetermined depth. Once this depth is reached, the tool and workpiece are then rolled together in a predetermined relative rolling motion, known as the generating roll, as though the workpiece were rotating in mesh with a theoretical generating gear, the teeth of the theoretical generating gear being represented by the stock removing surfaces of the tool. The profile shape of the tooth is formed by relative motion of the tool and workpiece during the generating roll.
In generating face hobbing processes, the tool and workpiece rotate in a timed relationship and the tool is fed to depth thereby forming all tooth slots in a single plunge of the tool. After full depth is reached, the generating roll is commenced.
Non-generating processes, either intermittent indexing or continuous indexing, are those in which the profile shape of a tooth on a workpiece is produced directly from the profile shape on the tool. The tool is fed into the workpiece and the profile shape on the tool is imparted to the workpiece. While no generating roll is employed, the concept of a theoretical generating gear in the form of a theoretical “crown gear” is applicable in non-generating processes. The crown gear is that theoretical gear whose tooth surfaces are complementary with the tooth surfaces of the workpiece in non-generating processes. Therefore, the cutting blades on the tool represent the teeth of the theoretical crown gear when forming the tooth surfaces on the non-generated workpiece.
The relationship between the workpiece and generating gear can be defined by a group of parameters known as basic machine settings. These basic settings communicate a sense of size and proportion regarding the generating gear and the workpiece and provide a common starting point for gear design thus unifying design procedures among many models of machines. The basic settings totally describe the relative positioning between the tool and workpiece at any instant.
Basic machine settings for forming gears are known in the art and may be defined as follows:                1. cradle angle (q) which defines the angular position of the tool about the cradle axis;        2. radial setting (S) which is the distance between the cradle axis and the tool axis;        3. swivel angle (j) which defines the orientation of the tool axis relative to a fixed reference on the cradle;        4. tilt angle (i) which defines the angle between the cradle axis and the tool axis;        5. root angle (γm) which sets forth the orientation of the work support relative to the cradle axis;        6. center-to-back or head setting (Xp) which is a distance along the work axis from the apparent intersection of the work and cradle axes to a point located a fixed distance from the workpiece;        7. work offset (Em) which defines the distance between the work axis and the cradle axis;        8. sliding base (Xb) which is the distance from the machine center to the apparent intersection of the work and cradle axes;        9. rotational position of the workpiece (ωw);        10. rotational position of the tool (ωt), for face hobbing;        11. ratio-of-roll (Ra) between cradle rotation and workpiece rotation, for generating.        
In conventional gear forming machines, the cradle angle, workpiece rotation and tool rotation change during generation while the other settings generally remain fixed. Two notable exceptions to this are helical motion which involves motion of the sliding base, Xb, and vertical motion which is motion on the work offset direction, Em.
The conventional mechanical machine meets the concept of the theoretical basic machine since nearly all machine settings correspond to theoretical basic settings. Such a machine may be illustrated by FIG. 2. In the mechanical machine, the basic setting for the radial, S, is controlled by an angular machine setting known as the eccentric angle.
Generating and non-generating processes are usually carried out on conventional mechanical gear generating machines or on multi-axis computer controlled (e.g. CNC) gear generating machines (such machines also being known as “free-form” machines). Conventional mechanical gear generating machines for producing bevel gears comprise a work support mechanism and a cradle mechanism. During a generating process, the cradle carries a circular tool along a circular path about an axis known as the cradle axis. This is known as the generating roll or cradle roll. The cradle represents the body of the theoretical generating gear and the cradle axis corresponds to the axis of the theoretical generating gear. The tool represents one or more teeth on the generating gear. The work support orients a workpiece relative to the cradle and rotates it at a specified ratio to the cradle rotation. Traditionally, conventional mechanical cradle-style bevel gear generating machines are usually equipped with a series of linear and angular scales (i.e. settings) which assist the operator in accurately locating the various machine components in their proper positions.
In multi-axis computer controlled gear generating machines, such as those disclosed by U.S. Pat. Nos. 4,981,402; 6,669,415 and 6,712,566, the disclosures of which are hereby incorporated by reference, movement of a tool relative to a workpiece along or about multiple machine axes (e.g. 5 or 6) can perform the cycle of movements including the kinematical relationship of the work and tool in the manner the same (or nearly the same) as that performed to generate a bevel gear in a conventional machine process utilizing a known face mill cutter or grinding wheel.
It has generally become the practice in the art to utilize the same input parameters (e.g. machine settings) as a conventional mechanical cradle-style gear generating machine for multi-axis computer controlled gear generating machines having a different number and/or configuration of axes. In other words, the positions of the tool and workpiece axes in the coordinate system of a conventional mechanical cradle-style bevel gear generating machine are transformed into the alternative coordinate system of the multi-axis computer controlled gear generating machine. Examples of such transformations can be found in the above referenced U.S. Pat. Nos. 4,981,402; 6,669,415 and 6,712,566, the disclosures of which are hereby incorporated by reference.
In the design of blanks for spiral bevel and hypoid gears, the face cone is defined as an imaginary cone tangent to the tops of the teeth. Traditionally, the face cone element or generatrix is a straight line intersecting the face cone apex. On the other hand, the tooth root lines are traced by the blade tips and normally are curved lines due to the facts that (1) the teeth are generated by a tilted circular cutter head and (2) for bevel gears that are generated with higher order modifications, the root lines are generally higher-order curves. See FIG. 1(a). As a result, the tooth top geometry may not fit the mating member's real root shape, providing an uneven tooth root-tip clearance. Additionally, in some cases, root-tip interference such as corner or edge contact, may be observed. FIG. 1(b) illustrates an example of corner interference and edge contact on the gear root surfaces and the pinion tip where highly concentrated contact stress may be present.
To avoid the root-tip interference, it is known to shorten the pinion teeth, as shown in FIG. 1(c), which results in reduced working area of the tooth surfaces and the contact ratio. Alternatively, to avoid interference, it is disclosed in U.S. Pat. No. 4,308,760 that the top of a tooth be formed such that contact with the tooth root of a mating member is prevented. The form of a tooth top may match the form of the root portion of a mating gear member. It is also known from U.S. Pat. No. 1,390,414 to provide teeth that are convex-shaped (tooth tops and tooth roots) in the longitudinal direction in order to decrease contact bending stresses.
The present invention proposes a method to modify the form of the face cone element in order to make the tooth tip line of one member of a gear pair fit the corresponding tooth root line of the mating member of the gear pair.